Elements of Stream Calculus (An Extensive Exercise in Coinduction)

Elements of stream calculus (an extensive exercise in coinduction) CWI's research has a theme-oriented structure and is grouped into four clusters. Listed below are the names of the clusters and in parentheses their acronyms. ABSTRACT Based on the presence of a nal coalgebra structure on the set of streams innnite sequences of real numbers, a coinductive calculus of streams is developed. The main ingredient is the notion of stream derivative, with which both coinductive proofs and deenitions can be formulated. In close analogy to classical analysis, the latter are presented as behavioural diierential equations. A number of applications of the calculus are presented, including diierence equations, analytical diierential equations, continued fractions, and some problems from discrete mathematics and combinatorics. 1 Contents 1 Introduction 3 2 Streams and coinduction 4 3 Behavioural diierential equations 6 4 Basic stream calculus 8 5 Solving behavioural diierential equations 17 6 Application: solving diierence equations 19 7 Solving quadratic equations in stream calculus 21 8 Shuue product and shuue inverse 24 9 Application: a d i v ergent recurrence 27 10 Comparing convolution product and shuue product 28 11 Application: a generalised Euler formula 31 12 Application: solving analytical diierential equations 32 13 Weighted stream automata 37 14 Multivariate streams 43 15 Multivariate stream calculus 45 16 Weighted automata for multivariate streams 46 17 Application: coinductive counting 47 18 Discussion and related work 50 19 Appendix 52